Study on the application of rigid body dynamics in the traffic accident reconstruction. Ming Ni

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1 Applied Mechanics and Materials Submitted: ISSN: , Vol. 707, pp Revised: doi: / Accepted: Trans Tech Publications, Switzerland Online: Study on the application of rigid body dynamics in the traffic accident reconstruction Ming Ni 1 Department of Physics and Technology, Kunming University, Kunming ,China a kmxynm@163.com Keywords: accident; rigid body; dynamics; collision damage. Abstract. The analysis of road vehicle collision accidents reconstruction is a hot automotive passive safety subject. When reconstruct the vehicle collision accidents, start from the final position of the vehicle, and build the back projections by using the kinematic and dynamic models, according to the relevant theory, experimental data. namely: the after collision stage - Collision stage - pre-crash phase to reproduce the actual situation of the entire accident process in time and space. Thus the technical state of vehicle accidents before the collision can be known, and the causes of the accidents can be further analyzed. In this paper, some aspects related to rigid body kinematics and rigid body dynamics will be studied. 1. Introduction With the rapid development of the economy, the automotive industry and road transport have also been rapidly developed. Road traffic accidents occur frequently, which not only brings a lot of inconvenience to road traffic management but also threatens people's lives and property. In many road accidents, vehicle collisions are the most serious and dangerous also the largest. So having a comprehensive, systematic study of the car's collision is becoming the world's most pressing subject. The whole process of the collision of moving vehicle is inseparable from the role of the force. So in order to better analyze vehicle collisions and reproduce this process, a lot of mechanical knowledge should be required. This paper studies the vehicle accident reconstruction mechanical problems, laying the foundation of further research. 2. Rigid body kinematics The two phases before and after the car collision can be approximated as a rigid body in the plane motion. The rigid position can be described by two coordinate different points on the body or in a rigid coordinate and angle. The trajectory of a rigid body can be described by its position changing with time centroid. Figure 2.1 Rigid planar motion Figure 2.2 At any moment instantaneous pole The rigid body rotation center M is defined by Euler theorem : Any changes of plane motion of rigid body position can be achieved by rotating around a fixed axis; the fixed axis is perpendicular to All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, USA-19/09/16,01:51:25)

2 Applied Mechanics and Materials Vol the plane of motion of a rigid body,whose intersection with the movement plane is called the public pole M. Any two rigid bodies A and B or their connection AB can describe the plane motion of a rigid body, shown in Figure 2.1. If there is a common rigid pole M, it is on the midperpendicular of displacement connection within the interval t; empathy M is certainly on the midperpendicular of Line formed by rotating the angle to of. So the point is at the intersection of the midperpendicular of Line and Line. The pole M is not only at the intersection of the Line and Line but also on the normal of the movement trajectory between A and B. The intersection of the normal of the movement trajectory between the two points is also called an instantaneous pole Mm. In the traffic accident reconstruction, the sense of Mm can be described as: at a time before a collision accident, There are at least the trcks of tires and A and B. Thus, the tire intersection point of the normal between two and points is the instantaneous pole of the vehicle, as shown in Figure 2.2. The trajectory of which changes with time in the fixed plane is defined as stagnation point of the trajectory, while the the instantaneous pole fixed with the plane of the rigid body is called the motion pole trajectory. By definition, this stagnation point trajectory and the trajectory of the two curves are at all the time. In other words, the pole trajectory is moving along the trajectory of stagnation. It is thought that any plane motion is formed by the stagnation point of a curve in the same fixed track. Rigid body motion is generally expressed by rotating and parallel movement. The infinitesimal motion of any point P in the rigid body is shown as in Figure 2.3 (2-1) In the formula is the radius vector; is the radius vector of the reference point; is angular displacement. Formula (2-1) derivative was (2-2) is speed; is the speed of the reference point ; is the angular velocity of the rigid body Formula (2-2)shows The state of motion of a rigid body can be completely described by the speed of the reference point and the angular velocity of the rigid body. It has nothing to do with selecting the reference point. Plane motion is a special case of rigid body motion. When selecting centroid as a reference point (as shown in Figure 2.4),there (2-3) Figure 2.3 Translation and rotation if the rigid body Figure 2.4 The velocity of the rigid body 3.Dynamics equations of the collision of the rigid body In the principle,there are two methods to carried out the traffic accident reconstruction, namely the collision force calculation and the force calculation. Calculation is performed for solving differential equations of motion. Centroid theorem: If the quality of the elements is considered as is shown in Figure 3.1, the centroid vectors per is set, then according to Newton's Second Law shows: (3-1)

3 414 Advanced Research on Materials, Chemistry and Informatization IV Assuming dm is the time constant of the above equation was derived: (3-2) The impulse theorem is introducted (3-4) (3-3) Both sides are Differentiated (3-5) So the following theorems are available by the centroid motion: (3-6) The formula (3-6) shows that the motion of the object is like all external force in its centroid. Therefore the movement of an object can be considered as both the motion of its centroid and a particle, too. Figure 3.1 Plane theory of momentum Figure 3.2 Mass momentum plane moments Momentum Theorem: By the formula (3-5) and then by arm length away, integrating the entire object can be derived: If the shear stress is ignored, the resultant moment vector on the object can be got: (3-7) The moment is available of momentum theorem rrom the above two equations: (3-8) The formula (3-8) shows that the external torque acts on an object should be equal to the first derivative of momentum moment of time, and the moment of momentum is equal to the product of the angular velocity vector and the moment of inertia. Planar rigid body dynamics equations: As Figure 3.2 shown, the dynamics equations for plane of the rigid body In the formula:: is the x direction external force, is the y direction external force, is the moment relative to Point S. 4. The energy loss of the rigid body when colliding The form of the vehicle collision is similar to a elastic-plastic collision. The body size of the extent of recovery can be represented by the elastic coefficient of restitution after the collision. Elastic

4 Applied Mechanics and Materials Vol coefficient of restitution e is defined as the ratio of the instantaneous speed difference the moment the two objects separate from and contact with each other. Namely: Vehicle collision will produce varying degrees of elastic-plastic deformation in the contact area of the two cars, resulting in mechanical energy not conserved before and after the collision. In a car and car frontal crash accident, the collision can be equivalent to that aginst the wall, the mechanical model is shown in Figure 4.1 Figure 4.1 Mechanical model car frontal crash According to Hooke's Law, the relation between the elastic force F of the spring and the elongation of the spring or the displacement x of the object starting from the balance position is: In the formula: k is the spring stiffness. Set According to Newton's Second Law, the car's acceleration a during the deformation process is: The mechanical model shows that the effective spring stiffness coefficient between the two vehicles is: because,if set,then: In the formula: c is the spring stiffness per unit of mass of the vehicle body of the front portion of the vehicle regardless of the value of about / m. Therefore, the energy loss due to plastic deformation of the collision is: (4-1) In the formula: is the difference between the moving distance of and, Namely algebraic sum of the deformation of the two vehicles. In addition, the energy loss in the inelastic collision is: In an accident with personal injury,the collision can be considered fully plastic,namely Therefore, according to the formula (4-1), (4-2) get: (4-2) The deformation of the two vehicles are:

5 416 Advanced Research on Materials, Chemistry and Informatization IV The plastic deformation x and thecollision force F can be approximated as a linear relationship the expression for F = 98l00x Thus, the energy absorbed by the plastic deformation of the two vehicles are as follows: When a frontal collision of two vehicles happens, the deformation is inversely proportional to the vehicle mass and the impact energy absorbed in the collision is inversely proportional to the square of the mass of the vehicle. Therefore, the lighter the car crash in the quality of the more serious damage, the greater the occupant casualties in a collision accident. 5. Conclusion Based on the reasonable analysis of the collision, the vehicle study, some reasonable assumes, the dynamics equations of rigid body,centroid theorem, moment of momentum theorem and the kinetic energy theorem, the collision theory model is constucted according to the type of accident collision, and energy absorption is derived after vehicle collision. In actual traffic accidents, the lighter the car crash in the quality of the more serious damage, the greater the occupant casualties.the reason is that the energy absorption is inversely proportional to the square of the mass of the vehicle the moment the collision happens. References [1] Hermann Steffan,B.C.Geile.A New Approach to Occupant Simulation through the Coupling of PC-Crash and MADYMO.SAE Paper.No :l-5 [2] Bruce Hongola,Cheng-Yao Chan.Simulation and Animation Tools for Analysis of Vehicle Collision: SMAC(Simulation Model Automobile Collisions)and Carmma (Simulation Animations).California PATH Working Paper.ISSN: :3-5 [3] Cheng Yao Chan.Studies of Vehicle Coll isions A Documentation of the Simulation Codes:SMAC(Simulation Model of Automobile Collisions).California PATH Working Paper.ISSN :2-7 [4] Mchenry B. G., Mchenry R. R. SMAC97 - refinement of the Collision algorithm.sae Paper.No :l-3 [5] Richard P.Howard,John Boomer,Cleve Bare.Vehicle Restitution Response in Low [6] Mchenry B.G.,Mchenry R.R.CRASH 97-refinement of the Trajectory Solution Procedure.SAE Paper.No :2-4 [7] James A.Neptune,James E.Flynn.Impact Analysi s Based upon the CRASH Damage Algorithm.SAE paper.no :1-2 [8] Kenneth.Campbell.Energy Basis for Collision Severity.SAE Paper.No :l-4 [9] Mchenry.R.R.Mchenry.B.G.A.Revised Damage Analysis Procedure For The CRASH Computer Program.SAE Paper.No :l-2 [10] Marcus Hiemer,Sebastian Lehr,Uwe Kiencke.A Fuzzy System to Determine the Vehicle Yaw Angle.SAE Paper.No :1-9 [11] Michael S.Varat,Stein E.Husher.Crash Pulse Modeling for Vehicle Safety Research.The 18 PESV Paper.No :1-5